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Symbols

Classical Electrodynamics

{Lo

p

Xe

Xm

dielectric constant; permittivity

dielectric constant of vacuum; ::: 8.854· 10-12 As/Vm

relative permittivity

(electric) conductivity; [1 I .om]

electric conductivity; [I/.om]

thermal conductivity; [W Im·K]

permeability

permeability of vacuum; 411' . 10-7 VsI Am ::: 8.854· 10-12

Vs/Am relative permeability

wave length

circular cutoff frequency

scalar potential

complex scalar potential

charge density; [As/m3 ]

dielectric susceptibility

magnetic susceptibility

354 Symbols

C

Co

A

B

B

D

D

e

E

E'

E

Ie

H

H'

H

J

l

JE

JK

JL

J w

k

speed of light

vacuum speed of light; c = 2.99792458· 108m/s

vector potential

magnetic flux; [TJ=[Vsjm2]

phasor; complex amplitude of B(r, t) = Re(B(r)eiwt ) in complex notation for time-harmonic magnetic flux

electric flux; [C/m2] = [As/m2]

phasor; complex amplitude of D(r, t) = Re(D(r)eiwt ) in complex notation for time-harmonic electric flux

electron charge; e = 1,60· 10-19 As

electric field strength; [V jm]

normalized electric field strength

phasor; complex amplitude of E(r,t) = Re(E(r)eiwt ) in complex notation for time-harmonic electric field

cutoff frequency

magnetic field strength; [Aim]

normalized magnetic field strength

phasor; complex amplitude of H(r, t) = Re(H(r)eiwt ) in complex notation for time-harmonic magnetic field

current density; [A/m2]

phasor; complex amplitude of J(r,t) = Re(~(r)eiwt) in complex notation for time-harmonic current density

impressed current density

convection current density

conduction current density

thermal flux; [W 1m2]

Boltzmann constant; k = 1,38.10-23 J /K

L typical length of a setup

M magnetization

p polarization density

S Poynting vector; [W 1m2 ]

t time

T temperature

v velocity vector

u energy as function of temperature

W heat source density

We electric energy density

Wm magnetic energy density

Yo admittance

Zo wave impedance

Mode Matching Technique

Cmn

resonant frequency

coupling matrix element

weighted coupling matrix element

coupling matrix

coupling matrix

identity matrix

field eigenfunction of the electric field

Symbols 355

356

hn

In

I

M

N

T(E)

T(H)

Un

U

Ws

Symbols

field eigenfunction of the magnetic field

current amplitude (Fourier coefficient)

vector holding the current amplitudes

number of eigenmodes

number of eigenmodes

cross-sectional eigenfunction of Ez-waves

cross-sectional eigenfunction of Hz-waves

voltage amplitude (Fourier coefficient)

vector holding the voltage amplitudes

stored energy

Finite Element Method

<Pi

f

H(curl; il)

H(div; il)

r

u

basis function

right-hand side of differential equation

function space in edge element formulation

function space in edge element formulation

interaction coefficients

linear differential operator

residual

unknown field; solution function

coefficients in approximation to u; nodal values

Symbols 357

Finite Integration Technique

P E,i

Pm

p. -~

b

c

d

e

(G,G)

G

G

i, j, k

discrete electric scalar potential

discrete magnetic scalar potential

magnetic charges (!non-physical auxiliary quantity!)

discrete complex scalar potential (EQS)

elementary area

vector holding the magnetic (grid-)fl.ux

magnetic (grid-)fl.ux

discrete curl operator

discrete curl operator on dual grid

vector holding the electric (grid-)fl.ux

electric (grid-)fl.ux

discrete material matrices

vector holding the electric (grid-) voltage

electric (grid-)voltage

grid duplet

FIT-grid

dual FIT-grid

magnetic (grid-) voltage

homogeneous part of magnetic field

inhomogeneous part of magnetic field

index of lexicographic numbering (in U-, V-, w-direction)

358 Symbols

J,J,K,N

qi

s

maximal index of the grid (in u-, v-, w-direction resp. total number of grid points)

total electric (grid-) current

elementary line

grid point

discrete operators corresponding to partial derivatives in x, y and z on dual-orthogonal grids

discrete charges

discrete divergence operator

discrete divergence operator on dual grid

discrete gradient operator

discrete gradient operator on dual grid

elementary volume; FIT-cell

Numerical Treatment of Linear Systems

parameter on S in Axelsson's method

... in cg-algorithm

... in cg-algorithm

cycling parameter in multigrid methods

~2(A) condition number of matrix A

relaxation parameter of the Kaczmarz algorithm

eigenvalue of system matrix

number of relaxations on grid G k

convergence factor of Theta-component

p(M)

fJ(PdM))

Pk

e

w

A

b

c

Symbols 359

smoothing factor

iteration function

BiCG-polynomial relating residual in step k to initial residual nearby BiCG-polynomial

BiCG-polynomial relating search direction in step k to initial residual spectral radius of matrix M

virtual spectral radius of matrix M

parameter in Chebyshev iteration

acceleration parameter

Fourier component of error function

relaxation parameter

system matrix

system matrix on grid with step size h

system matrix on grid Gt

element in row i and column j of the system matrix A

right hand side

right hand side on grid with step size h

right hand side on grid Gt

preconditioning matrix

Chebyshev polynomial of degree k

defect on grid with step size h

defect on grid Gt

360 Symbols

D

ei

Gn

Gh

GH

IH h

11- 1 1

Ih H

ILl

-I 11- 1

k

k' Km

L

M

n

P

Pk

Pk

P

Pk

Pk

diagonal matrix

i-th unit vector

system matrix of transformed system in Axelsson's method grid with step size h (fine grid)

grid with step size H (coarse grid)

restriction from fine to coarse grid

restriction from fine to coarse grid

interpolation from coarse to fine grid

interpolation from coarse to fine grid

interpolation from coarse to fine grid in FMG

wave number

m-th Krylov subspace

lower triangular matrix in Gaussian elimination

iteration matrix

matrix dimension

permutation matrix in Gaussian elimination

search direction in step k of cg-algorithm

pseudo-search direction

orthogonal projection

polynomial of degree k

projection on Vk

R

s

s

u

u

v

Vn,m

x

Xt

Symbols 361

residual vector after k iterations

pseudo-residual

real part of complex system matrix

shift-factor for eigenvalue k2 in multigrid algorithm

imaginary part of complex system matrix

tridiagonal matrix in Lanczos algorithm

real part of right hand side (Axelsson's method)

upper triangular matrix in Gaussian elimination

linear subspace

imaginary part of right hand side (Axelsson's method)

error on grid with step size h

error on grid Gl

coefficient in linear combination of approximate solutions after k Chebyshev iterations

error of approximation to solution on two-dimensional grid

affine subspace

weighting factor for under-interpolation in multigrid algorithm

affine subspace

solution vector resp. its real part (Axelsson's method)

solution vector on grid with step size h

solution vector on grid Gt

approximation to solution vector on grid with step size h

approximation to solution vector on grid Gt

362

Xl

X·

X·

Xk

X(k)

(k) Xi

Y

Yk

Symbols

initial approximation to solution vector on grid Gl (FMG)

fixpoint

exact solution

approximation to solution vector after k iterations

approximation to solution vector after k iterations

i-th component of k-th iterative approximation

complex part of solution vector (Axelsson's method)

approximation to solution vector after k iterations of Chebyshev algorithm

Index

O-mode, 289, 300, 311 27r /3-mode, 253, 255, 258, 282, 323 36-cell experiment, 295-302, 304-315,

319 7r-mode, 253, 255, 289, 300, 311 7r /2-mode, 253, 255 e+ e- -physics, see electron-positron

physics * operator, see Hodge operator 3 dB waveguide coupler, 187, 237

A-conjugate, see A-orthogonal A-norm, 100 A-orthogonal - basis, 101, 102 - projection, 101, 102 - vectors, 103 ABC , see absorbing boundary

condition absorbing boundary condition, 30 accelerating - gradient, 246, 258 - - loaded, 260 -- SBLC, 265 - mode, 253, 255, 282 - section, 243 - structure, 248, 253, 322

aperiodic, 258-264 constant gradient, 246, 255, 258-262, 265, 270, 278, 280 constant impedance, 258, 278 detuned, 258, 262-264 loaded,259 periodic, 257-258 quasi-constant gradient, 261, 282 rotationally invariant, 276 SBLC, 265, 271 SLAC, 270 tapered, 258

- voltage, 255 acceleration of charged particles, 244,

250

accelerator, 243 acoustics, 57 adaptive strategy, 127 admittance, 21 algebraic multigrid, see AMG AMG, 129, 214 AMLl,155 amplitudes - current, 39 - inner, 41 - outer, 42 - voltage, 39 - wave, 41 analytical solution methods, 23 aperiodic acceleration structure,

258-264, 280, 311 - loss parameter curve, 287 approximation quality of matrix

inverse, 151 Arnoldi method, 115 asymptotic convergence factor, 94 attenuation, 258, 282 - parameter, 257 auxiliary transitions, 46 Axelsson's method, 158 Axelsson's reduction of a complex

linear system to real form, 156-161

B-Factory, 324 backward SOR, 95, 154 balancing transformation of magnetic

field, 143 basis function, 51 BCl,274 bead, 303, 310 beam - break-up, 270, 279 - - multibunch, 280 - - single bunch, 279 - dynamics, 246, 268-280 - instabilities, 268, 275, 279-280 - loading, 246, 259, 260

364 Index

- - fundamental theorem, 259 - - transient, 280 - matched, 270 - mismatched, 270 - optics, 268 - position monitor, 318 beam pipe mode, 309 BEM, 226, 280 Bernoulli's product ansatz, 28 Bessel's differential equation, 28 - general solution, 28 beta function, 269 betatron - motion, 269 - oscillation, 269 bi-orthogonalization, 106 BiCG, 109, 110, 190 BiCG polynomials, 111 BiCGCR,113 BiCGSTAB, 115-117, 178, 185, 187,

190 BiCGstab(l), 118 BiCGstab2, 118, 178, 185, 187 BINP, 248 black box solver, 100, 130, 190 BNS damping, 279 board, 223 Boltzmann constant, 15 bond wires, 189 bordering methods, 150 boundary condition - perfect electric, 42 - perfect magnetic, 42 boundary element method, see BEM boundary operator, 31 boundary value problems, 29-33 - absorbing boundary condition, 30 - open boundary condition, 30 - periodic boundary condition, 30 - potential theory, 29

Dirichlet problem, 29 first boundary value problem, 29 mixed boundary value problem, 29

-- Neumann problem, 29 - - Newton problem, 29 - - second boundary value problem, 29 - - third boundary value problem, 29 - waveguide boundary condition, 31 box schemes and FIT grids, 68 breakdown, 118 - in BiCe, 110 Brillouin diagram, 257 bunch, see particle bunch

- charge, 319 - compression, 248 - distance, 246 - Gaussian, 273 - length, 273 - train, 246, 270 -- SBLC, 265 bunch-to-bunch energy spread, 280

C-magnet, 168, 212 - nonlinear, 216 C-to-R method, 158 - with Chebyshev iteration, 160, 161 - with preconditioning, 158 calculation domain, 57 Cartesian coordinate grid, 57 cavity, 244 cell-to-cell phase advance, 257, 271,

282,311 centroid, 270 CERN, 247, 248 cg, 99-105, 161, 162, 169, 170, 207, 209,

214, 216, 219 - error estimate, 104 - rate of convergence, 105 - stopping criterion, 104 - with ILU(3) preconditioning, 224 CGNE,109 CGNR,109 CGS, 112, 115, 161, 178, 185, 187 CGS2, 113, 178, 185, 187 CGW-method, 154 chain matrix, 41 charge density, 12 charged particle, 243, 260 Chebyshev acceleration, 98 Chebyshev iteration, 96-99, 150, 160 - and SSOR, 95 - error estimate, 99 Chebyshev polynomials, 97 chip, 237 choke-mode cavity, 317 Cholesky-factor, 152 circuit breaker, 221 classification of electromagnetic fields,

22 CLIC, 248, 249 coarse grid, 126 - correction, 126 coarsening for high frequency problems,

147 coce, 111, 177, 181 - with Minimal Residual Smoothing

(COCG-MRS), 182

collective effects, 268 COM, 281, 290-295 complex matrices - indefinite, 184 - special properties, 76-77 - symmetric, 184 - symmetric positive stable, 177 complex scalar potential, 24, 72, 224 computational domain, 31 computational effort, 31, 164, 172, 178,

182, 187 - BiCG, 111 - BiCGSTAB, 118 - BiCGstab(l), 118 - CGS, 112 - CGS2, 113 - COCG, 111 - GCG-LS, 121 - GMRES, 116 - model problem, 164-168 - SCBiCG, 113 condition number, 104, 150, 164 conductive contamination, 226 conductivity, 12 conjugate, 99 Conjugate Gradient algorithm, see cg connection coefficients, 51 conservation of charges, 13 conservation of energy, 14 consistent discretization method, 56, 64 constant gradient structure, 246, 255,

258-262, 280, 309, 315, 319 constant impedance landings, 261, 282 constant impedance structure, 246, 258,

310 contaminated insulator, 180 continuity condition, 40, 54 continuity equation, 13 convergence - behaviour, 178, 180-182, 187, 189 - - dependence on material parameters,

182 - factor, 124 - speed, 92 cooling of cavities, 323 Coulomb gauge, 23 coupled - calculation, 220 - problem, 205 - temperature problems, 320-324 coupled circuit model, 280 - verification, 287 coupling

- factor, 303 - impedance, 276 - matrix, 40 critical voltage, 225 cross-talk, 189, 237 cryostat, 323

Index 365

cumulative beam instability, 270, 280 cup, 250, 262 Curl-Curl Equation, 75, 234 current - amplitude, 39 - density, 12 - - conduction rv, 12 -- convection rv, 12 -- impressed rv, 12 - sensor, 174, 213 cut-off - diagram, 289 - number, 250 - tubes, 304 cylindrically symmetric structures, 36

DAE,233 damper, 271 damping, 244, 257, 319 - constant, 258 - factor, 258, 311 - global, 317-318 - local, 315 - strong, 311 - weak, 311 dark currents, 249 decay time, 273, 275 decoupling by differentiation, 25 defect, 125 - equation, 125 delta function, 271 design trajectory, 269 DESY, 248 determinant - after LV decomposition, 89 detuned structure, 258, 262-264 detuning, 280, 318 dielectric - constant, 12 - susceptibility, 12 differential-algebraic equations, 233 diffusion constant, 12 dipole component, 276 dipole modes, 38, 258, 260, 278, 283,

305,309 - strongly interacting, 287 direct Lanczos algorithm, 107 directional coupler, 237

366 Index

Dirichlet - (boundary value) problem, 29 - boundary condition, 212, 272 discharge, 225 - voltage, 225 discrete - charges, 61 - curl operator, 60, 63 - divergence operator, 63 - gradient operator, 63 discretization methods, 35 - finite difference method, see FD - finite element method, see FEM - finite integration technique, see FIT - finite volume method, 56 dispersion - curve, 251, 253 - - for periodic structures, 257 - function, 269 - relation, 250 displacement current, 225 dissipative heating, 320 divergence, 268 dual - FIT grid, 60 - - dual-orthogonal, 60, 61 - - for uniform coarsening, 134 - - special properties for dual-

orthogonal grids, 66 - formulation, 54 - grid, 60 - problem, 53 duality methods, 52 duality theory, 52

Ez-waves, 27 eddy currents, 320 edge, 54 - condition, 44, 46 - element formulation, 54, 55 effective emittance, 270 eigenfrequency, 244 eigenfunctions, 32 - cross-sectional eigenfunctions of E z -

and Hz-waves, 29 - expansion, 32 - field eigenfunctions, 38 - modes, 38 eigenvalue, 32 - distribution, 184 - of E z- and Hz-waves (Helmholtz

equation for circular cylindrical waveguides), 29

elastodynamics, 57 electric - (grid) current, 61 - (grid-)voltage, 58 - charge density, 12 - conductivity, 12 - displacement, 11 - field, 220, 227, 237, 239, 244, 276,

300 - - static, 207 - field strength, 11 - flux, 11, 61 - scalar potential, 23, 24 electro-quasistatic potential, 178, 227 electro-quasistatics (EQS), 17, 160,

177, 205, 224-233 - complex scalar potential, 24 - fundamental equations, 18 - with FIT, 72-73 electro-thermal behaviour, 320 electrode shape, 227 electrolytic partial discharge erosion,

226 electromagnetic fields, 11 - scattered, 272 electromagnetic waves, 21-22, 243 electron charge, 15 electron-positron collider, 247 electron-positron physics, 245 electrostatic model - vs. electro-quasistatic model, 227 electrostatic potential, 178, 207, 209 electrostatics, 16, 162, 205-211 - with FIT, 70 elementary - area, 57 - line, 57 - volumes, 57 elementary particle, 243 emittance, 268-270, 319 - effective, 270 - ellipse, 269 - growth, 270, 319 - multibunch, 270 - single bunch, 270 energy, 260, 318 - center of mass, 247 - conversion, 260 - density, 13, 324 -- electric, 13 - - magnetic, 13 - flux, 14 - gain, 258

- - in constant gradient structures, 260 - inner as function of temperature, 15 - kinetic, 244 - loss, 273, 274, 278, 322 - norm, 99, 100 - per cell - - in constant gradient structures, 315 - spread, 279 -- bunch-to-bunch, 280 - stored, 254, 278, 300 - supply, 244 - total, 302 - velocity, 251 epoxy-resin, 227 EQS, see electro-quasistatics equivalent circuit, 226, 290, 295 error estimate - cg, 104 - Chebyshev iteration, 99 error smoothing, 122, 123 exact breakdown, 108 excited fields, 273 expansion - into a series of eigenfunctions, 32 - multipole - - of wake potentials, 276 - orthonormal series, 33

FD, 35, 48, 280 FDTD,57 FEM, 48-56, 226, 280 - conforming approximation, 53 - degrees of freedom, 53 - edge element formulation, 54 - element properties, 50 . - external approximation, 53 - for Maxwell's equations, 54 - Hermite type elements, 53 - internal approximation, 53 - interpolation, 50 - Lagrange type elements, 53 - mixed, 54 - non-conforming approximation, 53 - system assembly, 50 - system solution, 50 - with Whitney forms, 54 field distribution - in 36-cell structure, 309 - in constant gradient structures, 260 field eigenfunctions, 38 fill-in, 90 filling time, 255, 315 filter

Index 367

- weighting eigenfunctions, 44 - weighting the coupling matrix, 44 fine grid, 126 finite difference method, see FD Finite Difference Time Domain, see

FDTD finite element method, see FEM finite elements, 48 - one-dimensional, 49 - three-dimensional, 49 - two-dimensional, 49 finite integration technique, see FIT first boundary value problem, 29 first law of thermodynamics, 15 FIT, 36, 56-76, 162, 178,205,226,244,

274, 281, 295, 323 - analogy with FD, 66 - cell, 57 - convergence of loss parameter, 289 - correspondence between discrete field

quantities and state variables, 66 - coupled thermal problems, 320 - discretization of integrals, 66 - grid, 57 - - correspondence to box schemes, 68 - - discrete charges, 61 -- dual, 60 -- dual-orthogonal, 60, 61 -- grid cells, 57 - - scalar potential, 62 - - state variables, 58 - - three-dimensional dual non-

orthogonal grids, 68 -- total electric current, 61 - - triangular grids, 68 - material matrices, 66 - properties, 64-65 - system matrices, 78-79 FIT grid - electric - - (grid-) voltage, 58 - magnetic - - gradient, 62 -- voltage, 61 Five-Point-Star, 94 fixed point, 92 - methods, 92 - - acceleration, 96 Floquet's theorem, 30, 257 FMC, 148 FMG approach, 129 force - exerted by wake fields, 274

368 Index

form factor, 304, 310 forward SOR, 154 Fourier coefficients, 39 Fourier series, 276 Fourier's law, 14 Fourier-Bessel series, 36, 37 - truncation, 44 frequency, 250 - accelerating - - tuning, 259 - excitation, 76, 79 - resonant, 42 Full MultiGrid, see FMG fundamental theorem of beam loading,

259

Galerkin approach, 50, 130 Galerkin's criterion, 51 Gauss algorithm, 161 Gauss-Seidel, 162 - algorithm, 126, 146 - method, 164 - - for restart, 172 - preconditioning, 154 Gaussian bunch, 273 Gaussian elimination, 89 - matrix inversion, 89 GCG-LS, 114, 120, 161 GCR,114 Generalized Conjugate Gradient, Least

Squares method, see GCG-LS Generalized Conjugate Residual

method, see GCR Generalized Minimal Residual, see

GMRES geometrical mode ratio, 44, 47 ghost modes, 55 Gibbs' phenomenon, 44 GMBACK,114 GMERR,114 GMRES, 114, 115 - restarted version, 116 GMRES(l), 116 grid - Cartesian coordinate, 57 - in multigrid scheme, 127 - non-coordinate, 57 - non-orthogonal, 57 - operator, 131 - pair, 57 grid-dependent eigenvalue shift,

140-141 group velocity, 251, 255, 257, 258, 282

- linear variation, 259

Hi-field, 71 Hz-waves, 27 Hall element, 219 hard materials, 16 harmonic - oscillation, 27, 244 -- with FIT, 74-76 - time dependence - - electro-quasistatic, 17 - - magneto-quasistatic, 18 heat - flow, 322 - load, 323 - source, 223, 324 - source density, 15 Helmholtz equation, 27, 75, 234 - homogeneous, 27 - inhomogeneous, 27 - scalar, 28 - - solution, 28 - vectorial, 28 Hermitian part, 77 high energy physics, 243, 247 high frequency - error component, 124 - fields, 74 - waves, 243 high voltage, 225 - insulators, 224-233 Higher Order Modes, see HOM Hodge operator, 55 HOM, 246, 250, 265, 279, 280, 319, 324 - 36-cell experiment, 295-302, 304-315 - computation, 281 - coupler, 315 - dampers, 288, 299 - damping, 247, 280, 315-319 - - by iris coating, 288 - in constant gradient structures,

288-295 - in quasi-constant gradient SBLC-like

structure, 281-288 - numerical analysis, 280-295 - trapped - - in aperiodic iris-loaded waveguides,

295-302, 304-315 HOM dampers, 310 hybrid methods, 116 - comparison, 121 hysteresis, 16

IC, 152,223

IC preconditioning, 162, 170 ICCG method, 174 ILU, 152, 161 - in accordance with matrix structure,

152 - with thresholds, 154 ILU )3(0), 153 ILU)3(3), 153 ILUw , 164 ILUw preconditioning, 167, 171, 174 ILU(O), 152 ILU(k), 153 ILUT(k), 154 image charge, 272 impedance, 149, 276, 295 - matrix, 43 implicit complex-valued split Jacobi

preconditioning, 182 incomplete Cholesky, see IC incomplete LU decomposition, see ILU induction law - discrete form, 60 inductive - coil, 320 - oven, 320 - soldering, 320 inductor, 320 injection error, 270 inner amplitudes, 41 input coupler, 247, 253, 257 insulator, 180, 225 interaction coefficients, 51 interaction region, 248 intermediate steps, 46 interpolation, 126 involutory matrix, 78 iris-loaded waveguide, 36 isotropic media, 16 iteration - error, 92 - function, 92 iterative methods - classical iteration methods, 91 - Krylov subspace methods, 91

Jacobi - matrix - - eigenvalues and eigenvectors, 163 - method, 92, 164 - preconditioned BiCGCR, 180, 184 - preconditioning, 170, 176 jitter, 270 JLC, 248, 249

Joule's energy, 320, 322 Joule's heat, 14

Kaczmarz algorithm, 146 Kaczmarz method, 95 KEK,248 kick factor, 278

Index 369

Krylov subspace, 87, 99, 102 - methods, 99-121,150,177, 181, 184,

205 -- comparison, 121

L-band,248 Lanczos algorithm, 99 - for linear systems, 107 - for non-hermitian matrices, 106 Lanczos-type algorithms, 105 landings, 261, 282 Laplace equation, 23 - on a rectangular domain, 32 Large Hadron Collider, 247 leap frog scheme, 57 least squares fit, 33 left-handed preconditioning, 164 LEP, 272 lexicographic ordering, 94 LHC, see Large Hadron Collider linear accelerator, 248 linear collider, 245 - beam dynamics, 268-280 - studies -- concluding remarks, 319 linear media, 16 linear system, 55 linewise relaxation, 94, 146 Liouville's Theorem, 270 loaded gradient, 260 local mode analysis, 131 longitudinal position, 268 Look-Ahead Lanczos, 108 Look-Ahead strategy, 118 Lorentz - contraction, 271 - factor, 250, 271 - force - - generated by wake fields, 274 - gauge, 25 loss factor, 278 loss parameter, 246, 260, 270, 278-279,

281, 285, 300, 309 - curve, 289 - - dependence on geometry, 294 - - dependence on tapering, 294 - - for constant gradient structure, 294

370 Index

- - for constant impedance structure, 294

- normalized, 278 loss parameter curve, 293 lossy cell, 318, 319 lossy sheets, 299 LU decomposition, 152 - with partial pivoting, 88 luminosity, 279, 319 lumped circuit - for resonance, 303 - model for constant gradient

structure, 290

MAFIA, 162, 274, 281, 289, 292, 295, 306, 311, 320

magnet - dipole, 268 - quadrupole, 268 - sextupole, 268 magnetic - (grid) flux, 59 - charges (auxiliary quantity), 71 - field strength, 11 - flux, 11 - - density, 212, 214, 215 - induction, 11 - scalar potential (auxiliary quantity),

71, 211 - susceptibility, 12 - vector potential, 23, 216 - voltage, 61 magnetization, 12, 302 magneto-quasistatics (MQS), 17, 18,

233-234 - with FIT, 73, 234 magneto-thermal behaviour, 320 magnetostatics, 16, 162, 211-216 - with FIT, 70-71 mass matrix, 55 matching, 36, 38 material - discrete approximation, 66 - dissipative, 253 - matrices, 66 material aging, 225 matrix condition, 212 Maxwell,l1 Maxwell Grid Equations, 63 - setup, 57 Maxwell's equations, 11-13,54 - Ampere's law - - discrete form, 62

- as an initial-boundary value problem, 21

- divergence equation - - discrete form, 62 - for quasistatic fields, 17 - - electro-quasistatics, 17 - - magneto-quasistatics, 18 - for stationary fields, 16-18 - for time-harmonic fields, 22, 27 - in differential form, 12 - in integral form, 12, 56 - induction law, 58 - - discrete form, 60 - non-existence of magnetic sources, 60 - - discrete form, 60 MBBU, see multibunch beam break-up measurement, 228 - experiments, 288, 294 - - 36-cell experiment, 295-302,

304-315 - non-resonant bead pull, 303 - resonant bead pull, 294, 302-304 - sampling frequency, 314 measuring velocity, 314 mesh generation, 48 method of integral equations, 35 method of moments, 35 method of separation, 26-29 MIC, 176 - preconditioning, 178 MICry, 162, 164 - preconditioning, 167, 171, 174 MICw (k),154 microchip, 189, 237 microwave systems, 243 MILU(k), 153 Minimal Residual smoothing, 113 MINRES, 114 mirror fields, 142 misalignment, 270 mixed boundary value problem, 29 mixed Finite Element Method (FEM),

54 mixed mode, 286 mode - 2~/3-, 253, 255, 258, 282 - ~-, 253, 255 - ~ /2-, 253, 255 - acceleration, 255, 282 - separation, 297, 309 mode matching technique, 35-47, 244,

265, 280, 295 - auxiliary transitions, 290

- convergence, 44-47, 289 - convergence criteria, 44 - - edge condition , 44 - - geometrical mode ratio, 44 - filter in, 290 - speed up by interpolation, 43 - system matrices, 77 model problem, 162 modes, 38, 244, 253, 257 - acceleration, see accelerating mode - dipole, see dipole modes - Higher Order, see HOM - monopole, 38, 278 - parasitic, see parasitic modes - quadrupole, 38 - trapped, see trapped modes modified IC, see MIC modified ILU, see MILU(k) modified ILUw preconditioning, 176 momentum, 273 momentum coordinates, 268 monopole component, 276 MQS, see magneto-quasistatics MR polynomials, 117 multibunch - beam break-up, 280 - dynamics - - longitudinal, 280 - - transverse, 280 - emittance, 270 - instabilities, 279-280, 319 - operation, 268, 270 multigrid, 214 - algorithm, 184 - for indefinite, nearly singular system,

138 - operator, 131 - scheme, 127 - techniques (MG), 121 multilevel preconditioner, 122, 155 multiply coupled oscillator model, see

COM

natural frequency, 42 near-breakdown, 108 nearly singular, 149 nested multi grid iteration, 129 Neumann - (boundary value) problem, 29 Neumann boundary condition, 212 Newton - (boundary value) problem, 29 NLC, 248, 249, 297, 318

Index 371

non-Hermitian, 100 non-resonant bead pull measurement,

303-305 non-stationary methods, 83, 85 non-symmetric and indefinite matrices,

184 nonlinear C-magnet, 216 normalization, 21

Ohmic losses, 253 open boundary condition, 30, 212 optimal relaxation parameter, 93, 164 - for SOR, 171 optimization of technical components,

243 ORTHO, 281, 289, 293, 295, 306 ORTHODIR, 114 orthogonal - Fourier series, 36 - functions, 31 - matrix, 78 - projection, 96, 101 ORTHOMIN, 120 orthonormal series expansion, 33 outer amplitudes, 42 over-relaxation, 92

Panofsky-Wenzel Theorem, 275, 276 parallel plate capacitor, 177 parasitic - fields, 244, 274 - modes, 245, 246, 250, 258, 260, 268,

284 partial - pivoting, 88 - SSOR preconditioning, 176, 184, 187 partially filled FIT cell, 136 particle bunch, 244, 246 pass band, 251, 258, 286 - overlap, 286 PBiCGCR, 182 PBiCGCR-MRS, 182 PCG,214 PCOCG,182 perfect - electric boundary condition, 42 - magnetic boundary condition, 42 perfectly conducting wall, 272, 273 periodic acceleration structure, 257-258 - loss parameter curve, 287 periodic boundary condition, 30, 323 permeability, 12 - nonlinear, 216 permittivity, 12

372 Index

permutation matrix, 88 perturbation theory, 279, 315 Petrov-Galerkin condition, 102, 110,

118 phase - advance, 253, 269 -- cell-to-cell, 257, 271, 282, 286, 294,

311 - angle, 260 - constant, 250 - ellipse, 269 - - invariance, 270 - space, 268 - - projection, 268 - velocity, 251, 257, 260 phasers, 18, 276 pick-up monitors, 288, 319 pivot element, 87 plate capacitor, 243 plug, 205 point charge, 271 - highly relativistic - - field of, 271 Poisson equation, 23 - complex (electro-quasistatics), 24 polarization, 11, 302 - density, 11 polynomial preconditioning, 155 polynomial representation - BiCG, 111 - BiCGSTAB, 117 - CGS, 112 - CGS2, 113 - COCG, 112 positive definite matrix, 77, 162 positive semi-definite matrix, 77 positive stable matrix, 77 potential equation, 23 potential theory, 23-25, 29 power, 257 - absorption, 315 - conversion, 258 - density, 14, 324 - flow, 253 - loss, 253 - rf-, 255 - supply, 253 power-loss method, 253, 323 Poynting vector, 14 Poynting's theorem, 14 preconditioned methods, 182 - cg, 161, 166, 214 - - on circular cylindrical grids, 175

- CGS, 187 - COCG, 184 preconditioner - SSOR,95 preconditioning, 150-156 - left-handed, 151 - of Krylov subspace methods, 190 - one-sided, 151 - right-handed, 151 - split, 151 principle of multigrid approach, 123 product ansatz of Bernoulli, 28 projection method, 110 propagation constant, 257 pseudo-residual, 110 PSSOR-BiCGSTAB, 189 PSSOR-CGS, 187, 189 PSSOR-TFQMR, 187, 189

QMR, 108, 118, 184 quadrupole component, 276 quality factor, 255, 259, 275, 295, 306,

318 Quasi Minimal Residual, see QMR quasi-constant gradient structure, 261,

282 quasi-symmetric matrix, 77 quasistatic approximation, 18 quasistatic fields, 17 - conditions for, 18-20 - electro-quasistatics, 17 - magneto-quasistatics, 17 - with FIT, 72-73, 234

rate of convergence - cg, 105 re-entrant corners, 148 real symmetric matrix, 162 Red-Black Gauss-Seidel, 94 red-black ordering, 94 Red-Black SOR, 94 reflection coefficient, 237 reflection matrix, 42 relative iteration error, 92 relative residual, 92, 169 relaxation, 92, 125 - for indefinite problems, 146 - optimal relaxation parameter, 93 residual, 51, 92 resistive wall wakes, 272 RESO, 281, 306 resonance, 244 resonant - bead formula, 302

- bead pull measurement, 294, 302-303 - cavity, 244 - frequency, 42, 244, 303, 306 - mode, 244 restart, 170 restriction, 126 rf deflection, 279 rf-load, 324 rf-window, 323 Ritz-Galerkin approximation, 101 Robbin's boundary condition, 31 robustness, 100 Rogowski profile, 227 rounding errors, 90 row scaling, 88 row-action procedure, 95

S-band, 245, 248, 249 S-band structure, 320 SBBU, see single bunch beam break-up SBLC, 245, 248, 249, 265-268, 271,

280, 297, 318-320, 323 - accelerating module, 266 - damped accelerating structure, 318 - main parameters, 265 - test facility, 318 - tunnel layout, 266 scalar potential, 23, 35 - for electro-quasistatics, 24 - for stationary current fields, 24 scalar potential function, 23 scalar wave equation, 25 scattering matrix, 38, 41-44, 80, 281 - concatenation, 41 SCBiCG(r, n), 113 SCBiCG(r, n) class, 177 search directions, 103 second boundary value problem, 29 semi-analytical methods, 35, 280 - method of integral equations, 35 - method of moments, 35 - mode matching technique, 35 semi-coarsening, 150 semiconductor, 220 sensor, 213, 214 separation equation, 29 separative plane, 43 SGS,95 SGS preconditioning, 154, 170 shunt impedance, 255, 259, 275, 306,

318 simple plate capacitor, 178 - with layer of water, 178

Index 373

simple test example with current excitation, 184

single bunch - beam break-up, 279 - dynamics - - longitudinal, 279 - - transverse, 279 - emittance, 270 - instabilities, 279-280, 319 singularity, 54 skew projection, 107 skew-Hermitian part, 77 skin depth, 253 SLAC,248 - two-mile accelerator, 270 Slater formula, 303 SLC, 247, 272 slow wave structure, 30, 250 slowly varying fields, 74, 225 smoothing factor, 124 smoothing procedure, 125 soft materials, 16 SOR, 93, 162, 169, 170, 216 - convergence, 93 - sensitivity to the relaxation

parameter, 171 - with optimal relaxation parameter,

164, 176 source field, 75 source-free field, 75 space harmonics, 257 spd matrix, 100, 162 spectral condition number, 159 spectral radius, 92 - of Jacobi matrix, 164 spectrum, 150 speed of light, 250 split preconditioner, 151, 154 spurious modes, 55 SSOR, 95, 164 - preconditioning, 154, 166, 171, 174 stabilized BiCG, 116-118 staggered grids, 57 stagnation, 185, 187 - in BiCGstab(2), 118 standing wave, 42 state variables, 58 - allocation, 58 - correspondence to discrete field

quantities, 66 - of a partially filled elementary area,

136 stationary

374 Index

- current fields, 16, 217-221 - - scalar potential, 24 - - with FIT, 71 - current problems, 162 - fields, 15-17 -- with FIT, 70-72 - heat conduction, 223-224 - temperature fields, 321 -- with FIT, 72 - temperature problems, 162 stationary methods, 83 stop band, 251 stopping criterion, 92 - for cg, 104 storage requirement - BiCG, 111 - BiCGSTAB, 118 - BiCGstab(l)2l+5 vectors, 118 - CGS, 112 - CGS2, 113 - COCG, 111 - GMRES, 116 - non-Hermitian Lanczos algorithm,

107 - of Gaussian elimination, 89 - of iterative methods, 91 storage ring, 243, 245, 247 stored energy, 43, 254 subdivision - for mode matching, 36 successive over-relaxation, see SOR superconductive accelerating structure,

248 superposition of waves, 42 surface charge, 272 symmetric Gauss-Seidel, see SGS symmetric matrix, 78 symmetric SOR, 95 symmetry plane, 212 SYMMLQ,107 synchrotron radiation, 245, 247

tapered, 258 TBA, see Two-Beam Accelerator TBCI,274 TEol-mode, 323 TE-waves,27 TEAM benchmark problem, 234 TEM-waves, 27 temperature, 14 - distribution, 224, 323, 324 - problems, 57, 223 TESLA, 248, 249, 323

TFQMR, 118, 119, 177, 178, 185, 187 thermal - conduction, 14 - conductivity, 14 - flux, 14 - load, 258 third boundary value problem, 29 three term recurrence, 87 three-dimensional dual non-orthogonal

FIT grids, 68 time-dependence - general time-dependence, 21 - harmonic time-dependence, 22 - - electro-quasistatic, 17 - - magneto-quasistatic, 18 time-dependent - fields, 239 -- with FIT, 74-76 time-harmonic - fields, 184, 234-239 - - electro-quasistatic, 17 - - magneto-quasistatic, 18 - oscillations, 22 TMolO mode, 244 TM-waves, 27 toroid electrode, 227 tracking, 270 transmission, 303 transmission matrix, 42 transport matrix, 270 Transpose-free Quasi-Minimal Residual,

see TFQMR transversal coordinates, 268 transversal waves, 27 - transverse electric, 27 - transverse magnetic, 27 transverse motion, 268 trapped modes, 43, 245, 247, 281,

286-302, 304-315, 319 - field pattern, 287 - geometry studies, 288-289 traveling wave, 42, 244 - structure, 249 - tube, 320 traveling wave tube - iris-loaded, 258 triangular FIT grids, 68 tuning, 259 Two-Beam Accelerator (TBA), 249 two-grid method, 126 two-grid operator, 131

ultra-relativistic particle, 244 under-interpolation, 145

under-relaxation, 93 uniform coarsening of a FIT grid, 134 unit cube, 162 URMEL,147 URMEL-I, 147, 149 URMEL-T, 278, 281, 288, 289, 295

V-cycle, 128, 148 vacuum system, 323 vector potential, 23, 25, 27, 35 vector wave equation, 25 velocity of light, 12 velocity sensor, 214 virtual spectral radius, 97 VLEPP, 248, 249 voltage, 255, 278 - amplitude, 39 - beam-induced, 260 - distribution (diagram), 283, 284, 297 - effective, 260 - generator-, 260

W-cycle, 128, 148 wake fields, 246, 249, 259, 268, 270-278,

319 - long range, 264, 275, 280 - - recoherence, 262 - long range forces, 277 -- HOM, 277 - longitudinal, 279 - resistive wall, 272 - short range, 271, 275, 279 - short range forces, 277 - transverse, 279 wake function, 277, 278 wake potential, 274-278 - m-pole component, 276 - longitudinal, 274 - - multi pole expansion, 276 - multi pole expansion, 276 - transverse, 274 - - multi pole expansion, 277 wall currents, 323

wall losses, 300, 320-322 water droplet, 227 wave - amplitude, 41 - impedance, 21 - length, 251 - number, 250 wave equation, 25 - for a damped wave, 26

Index 375

- - inhomogeneous equation for electric field, 26

wave front - reflection of, 273 waveguide, 237, 243, 323 - circular cylindrical, 27 -- iris-loaded, 36 - lossy termination, 324 - mode matching for cylindrical

waveguide, 39 - with load, 324 waveguide boundary condition, 31, 184,

324 waveguide coupler, 187 waves - in circular cylindrical waveguides, 27 - - TEM-waves, 27 - incident, 41 - reflected, 41 - standing, 42, 257 - superposition, 42 - transmitted, 41 - traveling, 42, 257 weak formulation, 54 weighted residual method, 50 Whitney forms, 54 Wiedemann-Franz' law, 15

X-band, 248

Young's Theorem, 93

Z-matrix, 43 Zwischenmediumsmethode, 46

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